Light groups of isomorphisms of Banach spaces and invariant LUR renormings

Leandro Antunes, Valentin Ferenczi, Sophie Grivaux and Christian Rosendal

Vol. 301 (2019), No. 1, 31–54

DOI: 10.2140/pjm.2019.301.31

Abstract

Megrelishvili (2001) defines light groups of isomorphisms of a Banach space as the groups on which the weak and strong operator topologies coincide, and proves that every bounded group of isomorphisms of Banach spaces with the point of continuity property (PCP) is light. We investigate this concept for isomorphism groups $G$ of classical Banach spaces $X$ without the PCP, specially isometry groups, and relate it to the existence of $G$-invariant LUR or strictly convex renormings of $X$.

Keywords

groups of isomorphisms of Banach spaces, isometry groups, LUR renormings, renormings of Banach spaces, light groups, invariant renormings

Mathematical Subject Classification 2010

Primary: 22F50, 46B03

Milestones

Received: 21 December 2017

Revised: 25 May 2018

Accepted: 26 September 2018

Published: 16 September 2019